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Chapter 3: Problem 3

Find the critical points of the given finction and then determine whether they are local maxima, local minima, or sadille points. $$f(x, y)=x^{2}+y^{2}+2 x y$$

Short Answer

Expert verified
The critical points are along \((x, -x)\), and their nature is indeterminate.

Step by step solution

01

Find Partial Derivatives

First, calculate the partial derivatives of the function with respect to \(x\) and \(y\). \[ f_x = \frac{\partial}{\partial x}(x^2 + y^2 + 2xy) = 2x + 2y \]\[ f_y = \frac{\partial}{\partial y}(x^2 + y^2 + 2xy) = 2y + 2x \]
02

Set Partial Derivatives to Zero

Set the partial derivatives equal to zero to find the critical points. \[ 2x + 2y = 0 \]\[ 2y + 2x = 0 \] Both equations are equivalent, thus simplifying to \(x + y = 0\).
03

Solve for the Critical Points

Solve \(x + y = 0\) to find the critical points. This gives \(y = -x\). Substitute \(y = -x\) into either of the partial derivative equations.Choose \(x = a, y = -a\). The general solution for \((x, y)\) is \((a, -a)\).
04

Calculate the Hessian Matrix

Calculate the second partial derivatives and form the Hessian matrix \[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = 2 \]\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = 2 \]\[ f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 2 \] The Hessian matrix is:\[H = \begin{bmatrix}2 & 2 \2 & 2 \\end{bmatrix}\]
05

Determine the Nature of Critical Points

To classify the critical points, evaluate the determinant of the Hessian matrix.\[ \det(H) = 2\cdot2 - 2\cdot2 = 4 - 4 = 0 \]Since the determinant is 0, we cannot conclusively determine the nature of the critical points; they could be saddle points, local maxima, or minima.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives are fundamental in analyzing functions of multiple variables. For a given function like the one in our exercise, a partial derivative shows how the function changes as one variable changes, while all other variables are held constant. In the function \(f(x, y) = x^2 + y^2 + 2xy\), we find the partial derivatives with respect to \(x\) and \(y\). This provides us with two equations: \(f_x = 2x + 2y\) and \(f_y = 2y + 2x\).
  • Understanding Partial Derivatives: Think of them as slices of a multi-dimensional pie. By focusing on one variable at a time, we can understand how each influences the function's outcome.
  • Identifying Critical Points: Setting these partial derivatives to zero helps us locate critical points, which are the potential sites for maxima, minima, or saddle points.
These derivatives not only help find critical points but also pave the way for deeper analysis using tools like the Hessian matrix.
Hessian Matrix
The Hessian matrix is a powerful tool in multivariable calculus for understanding the curvature of a function at a given point. It consists of second-order partial derivatives and provides information on whether a critical point is a local maximum, minimum, or saddle point.
  • Constructing the Hessian: For our function, the Hessian matrix is created by calculating \(f_{xx}, f_{yy},\) and \(f_{xy}\). In this case, it's \(H = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\).
  • Role of the Hessian: It serves as a multi-dimensional extension of the second derivative test used for single-variable functions.
In our example, computing the determinant of the Hessian helps in classifying the critical points, although a zero determinant indicates we might need further analysis.
Local Maxima and Minima
Local maxima and minima are points where the function takes on a highest or lowest value in a small surrounding area. Identifying these points is crucial as they represent key features of the function's landscape.
  • Role of the Hessian: If the determinant is positive and \(f_{xx} < 0\), the point is a local maximum. If \(f_{xx} > 0\), it鈥檚 a local minimum.
  • Challenges with Zero Determinant: In cases like our example where the determinant is zero, the test is inconclusive, suggesting the need for alternative assessments (e.g., analyzing eigenvalues).
Understanding how critical points relate to these local extremes assists in tasks like optimization and achieving insights into the function's behavior.
Saddle Points
Saddle points are a unique type of critical point where the function does not turn around in either a maximum or minimum sense. Instead, they represent points where the function curves up in one direction and down in another, resembling a saddle.
  • Identifying Saddles: The traditional signature of a saddle point is having a negative determinant for the Hessian, which indicates mixed curvatures.
  • Mixed Behavior: These points are neither peaks nor valleys but crossings where the function changes concavity.
In our function example, the zero determinant leaves the nature of \( (x, y) = (a, -a) \) undetermined, suggesting the necessity for more comprehensive explorations of paths in multi-dimensional spaces.

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Most popular questions from this chapter

Let \(f(x, y)=x^{2}+3 y^{2} .\) Find the maximum and minimum values of \(f\) subject to the given constraint (a) \(x^{2}+y^{2}=1\) (b) \(x^{2}+y^{2} \leq 1\)

Find the maximum and minimum values attained by \(f(x, y, z)=x y z\) on the unit ball \(x^{2}+y^{2}+z^{2} \leq 1\).

Let \(n\) be an integer greater than 2 and set \(f(x, y)=\) \(a x^{n}+c y^{n},\) where \(a c \neq 0 .\) Determine the nature of the critical points of \(f.\)

A rectangular box with no top is to have a surface area of \(16 \mathrm{m}^{2} .\) Find the dimensions that maximize its volume.

Denotes the unit disc. Let \(u\) be a harmonic function on \(D-\) that is, \(\nabla^{2} u=0\) on \(D \backslash \partial D-\) and be continuous on \(D\). Show that if \(u\) achicves its maximum value in \(D \backslash \partial D\), it also achicves it on \(\partial D .\) This is sometimes called the "weak maximum principle" for harmonic functions. [HINT: Consider \(\nabla^{2}\left(u+\varepsilon e^{x}\right), \varepsilon>0 .\) You can use the following fact, which is proved in more advanced texts: Given a sequence \(\left\\{\mathbf{p}_{n}\right\\}, n=1,2, \ldots,\) of points in a closed bounded set \(A\) in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\), there exists a point \(\mathbf{q}\) such that every neighborhood of \(q\) contains at least one member of \(\left.\left\\{\mathbf{p}_{n}\right\\} .\right]\)
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